Metric Diophantine approximation and 'absolutely friendly' measures

Abstract

Let $W(\p)$ denote the set of $\p$-well approximable points in $\R^d$ and let $K$ be a compact subset of $\R^d$ which supports a measure $μ$. In this short note, we show that if $μ$ is an `absolutely friendly' measure and a certain $μ$--volume sum converges then $μ(W(\p) \cap K) = 0$. The result obtained is in some sense analogous to the convergence part of Khintchines classical theorem in the theory of metric Diophantine approximation. The class of absolutely friendly measures is a subclass of the friendly measures introduced by D. Kleinbock, E. Lindenstrauss and B. Weiss (On fractal measures and Diophantine approximation) and includes measures supported on self similar sets satisfying the open set condition. We also obtain an upper bound result for the Hausdorff dimension of $W(\p) \cap K $.